400 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



may be written 



df df i-aaY'^ 



dt 



'V2nV a/2;^ {-a) 



2/3 



(30) 

 (31) 



(32) 



so that upon substitution from (29) and (31), (17) becomes 

 Fig. 5, which has been referred to above, shows 



O-a . /2eVa 

 'fcV 'kf- 



as a function of {fc/fa) as obtained from (32), and allows o-„ to be de- 

 termined easily. Using (20), the value of re+' is given by 



/ Tea , 



F 



-,.= -^^_,. = ,/_g-l) (33) 



where dg is the half-angle of the cathode (and hence the initial angle 

 which the path of a non-thermal edge electron makes with the axis). 

 We may write for 1/Fd 



1 V fe /d(-aY"\ 



Fo 4F 4(-aa)^/VV\rf(fc/r-) 7a 



(34) 



In Fig. 10 we plot —falFr, as a function of fjfa for easy evaluation of 

 re+' in (33). Taking the first derivative of (32) with respect to ^, we ob- 

 tain an expression for aJ. Using this in conjunction with (20) and (34) 

 we find 



0-+ = 



Y (r<^i + C2) 



I 



(35) 



where 





cira 



d{fc/f) 



/3 



and 



^-i/f. ft -(-''"/ 



(-a)2/3_ 



! 



Ci and C2 are plotted as functions of fc/fa in Fig. 11. 



(b) After choosing a specific value for r and evaluating K = rj/c/ . 



